Skip to main content
Log in

Abstract

In this paper, we establish energy estimates for Einstein vacuum equations in order to prove the evolutionary formation of black holes. The novelty of the paper is that we can completely avoid using rotation vector fields to prove the global existence of solution. More precisely, we use only canonical null vector fields as commutators to derive energy estimates at the level of the first derivatives of null curvature components. The special cancelations from the null structure of non-linear interaction yield the desired estimates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bieri, L., Zipser, N.: Extensions of the Stability Theorem of the Minkowski Space in General Relativity. Studies in Advanced Mathematics, Providence, RI: Amer. Math. Soc., Intre Press, 2009

  2. Christodoulou, D.: The Formation of Black Holes in General Relativity. Monographs in Mathematics, Zürich: European Mathematical Soc., 2009

  3. Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of Minkowski space. Princeton Mathematical Series 41, Princeton, NI: Princeton Univ Press, 1993

  4. Klainerman, S., Rodnianski, I.: On the Formation of Trapped Surfaces. http://arxiv.arg/abs/0912.5097v2 [gr-qc], 2009

  5. Klainerman, S., Rodnianski, I.: On emerging scarred surfaces for the Einstein vacuum equations. http://arxiv.arg/abs/1002.2656v1 [gr-qc], 2010

  6. Penrose R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 57–59 (1956)

    Article  MathSciNet  ADS  Google Scholar 

  7. Reiterer, M., Trubowitz, E.: Strongly focused gravitational waves. http://arxiv.arg/abs/0906.3812v2 [gr-qc], 2011

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pin Yu.

Additional information

Communicated by P. T. Chruściel

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yu, P. Energy Estimates and Gravitational Collapse. Commun. Math. Phys. 317, 273–316 (2013). https://doi.org/10.1007/s00220-012-1617-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1617-6

Keywords

Navigation